In “On the Phrase Potential Energy” (1867), William Rankine wrote, in response to a criticism by John Herschel:
In the course of an essay by John Herschel “On the Origin of Force,” which appeared some time ago in the Fortnightly Review, and has lately been republished in a volume, entitled Familiar Lectures on Scientific Subjects, the opinion is expressed that the phrase “potential energy” is unfortunate, inasmuch as it goes to substitute a truism for the announcement of a great dynamical fact” (Familiar Lectures, page 469).
There is here no question as to the reality of the class of relations amongst bodies to which that phrase is applied, nor as to any matter of fact concerning those relations, but as to the convenient and appropriate use of language. This is a sort of question in the discussion of which authority has much weight; and when an objection to the appropriateness of a term is made by an author who is not less eminent as a philosopher than as a man of science, and whose skill in the art of expressing scientific truth in clear language is almost unparalleled, it becomes the duty of those who use that term to examine carefully their grounds for doing so.
As the phrase “potential energy,” now so generally used by writers on physical subjects, was first proposed by myself in a paper “On the General Law of the Transformation of Energy,” | read before the Philosophical Society of Glasgow, on the 5th of January, 1853, I feel that the remark of John Herschel makes it incumbent upon me to explain the reasons which led me, after much consideration, to adopt that phrase for the purpose of denoting all those relations amongst bodies, or the parts of bodies, which consist in a power of doing work dependent on mutual configurations.
The kind of quantity now in question forms part of the subject of proposition 39[4] of Newton’s Principia; but it is there represented by the area of a figure, or by symbols only, and not designated by a name; and such is also the case in many subsequent mathematical writings.
The application of the word “force” to that kind of quantity is open to the objection, that when “force” is taken in the sense in which Newton defines “vis motrix”, the power of performing work is not simply force, but force multiplied by space.[5]
W = Fd
To make such an application of the word “force,” therefore, would have been to designate a product by the name properly belonging to one of its factors, and would have added to the confusion which has already arisen from the ambiguous employment of that word.
The word “power”, though at first sight it might seem very appropriate, was already used in mechanics in at least three different senses: viz., first, the power of an engine, meaning the rate at which it performs work, and being the product of force and space divided by time:
P = Fd/t
secondly, the power, in the sense of effort or pressure, which drives a machine; and thirdly, “mechanical powers,” meaning certain elementary machines. Thus, “power” was open to the same sort of objection with “force.”
About the beginning of the present century, the word “energy” had been substituted by Thomas Young [1807] for “vis viva”, to denote the capacity for performing work due to velocity:
Energy [kinetic] = vis viva = 1/2mv^2
and the application of the same word had at a more recent time been extended by William Thomson to capacity of any sort for performing work.[6] There can be no doubt that the word “energy” is specially suited for that purpose; for not only does the meaning to be expressed harmonize perfectly with the etymology of ενεργεια [energeia], but the word “energy” has never been used in precise scientific writings in a different sense; and thus the risk of ambiguity is avoided.
It appeared to me, therefore, that what remained to be done, was to qualify the noun “energy” by appropriate adjectives, so as to distinguish between energy of activity and energy of configuration. The well-known pair of antithetical adjectives, “actual” and “potential”, seemed exactly suited for that purpose; and I accordingly proposed the phrases “actual energy” and “potential energy”, in the paper [Rankine, 1853][1] to which I have referred.
I was encouraged to persevere in the use of those phrases, by the fact of their being immediately approved of and adopted by William Thomson; a fact to which I am disposed to ascribe, in a great measure, the rapid extension of their use in the course of a period so short in the history of science as fourteen years. I had also the satisfaction of receiving a very strong expression of approval from Baden Powell.
Until some years afterwards I was not aware of the fact, that the idea of a phrase equivalent to “potential energy,” in its purely mechanical sense, had been anticipated by Carnot, who, in an essay on machines in general, employed the term “force vive virtuelle” [virtual live force], of which “potential energy” might be supposed to be almost a literal translation. That coincidence shows how naturally the phrase “potential energy”, or something equivalent, occurs to one in search of words appropriate to denote that power of performing work which is due to configuration, and not to activity.
Having explained the reasons which led me to propose the use of the phrase “potential energy”, I have next to make some observations on the objection made by John Herschel to that phrase, that “it goes to substitute a truism for a great dynamical fact.”
It must be admitted that the use of the term “potential energy” tends to make the statement of the law of the conservation of energy wear, to a certain extent, the appearance of a truism. It seems to me, however, that such must always be the effect of denoting physical relations by words that are specially adapted to express the properties of those relations; or, what amounts virtually to the same thing, of drawing up precise and complete definitions of physical terms. Let A and B denote certain conceivable relations, and let them be precisely and completely defined; then, from the definitions follows the proposition, that A and B are related to each other in a certain way; and that proposition wears the appearance of a truism, and is virtually comprehended in the definitions. But it is not a bare truism; for when with the definitions are conjoined the two facts, ascertained by experiment and observation, that there are relations amongst real bodies corresponding to the definition of A, and that there are also relations amongst real bodies corresponding to the definition of B, the proposition as to relation between A and B becomes not a bare truism, but a physical fact. In the present case, for example, “actual energy” and “potential energy” are defined in such a way as to make the proposition: That what a body or a system of bodies gains in one form of energy through mutual actions, it loses in the other form — in other words, that the sum of actual and potential energies is “conserved” — follow from the definitions, so as to sound like a truism; but when it is proved by experiment and observation that there are relations amongst real bodies agreeing with the definitions of “actual energy” and “potential energy”, that which otherwise would be a truism becomes a fact.
A definition cannot be true or false; for it makes no assertion, but says, “let such a word or phrase be used in such a sense;” but it may be real or fantastic, according as the description contained in it corresponds, or not, to real objects and phenomena; and when, by the aid of experiment and observation, a set of definitions have been framed which possess reality, precision, and completeness, the investing of a physical fact with the appearance of a truism is often an unavoidable consequence of the use of the term so defined.
In the case of physical quantities in particular, the definition involves a rule for measuring the quantity; and the proof of the reality of the definition is the fact, that the application of the rule to the same quantity under different circumstances gives consistent results, which it would not do if the definition were fantastic; and hence the definitions of a set of physical quantities necessarily involve mathematical relations amongst those quantities, which, when expressed as propositions and compared with the definitions, wear the appearance of truisms, and are at the same time statements of fact.
In illustration of the foregoing principles, it may be pointed out that there is a certain set of definitions of the measurement of time, force, and mass, which reduce the laws of motion to the form of truisms, thus:
-Let “equal times” mean the times in which a moving body, under the influence of no force, describes equal spaces. This definition is proved to be real by the fact, that times which are equal when compared by means of the free motion of one body, are equal when compared by means of the free motion of any other body. If the definition were fantastic, times might be equal as measured by the free motion of one body, and unequal as measured by that of another.
-Let “force” mean a relation between a pair of bodies such that their relative velocity changes, or tends to change, in magnitude or direction, or both; and let “equal forces” mean those which act when equal changes of the relative velocity of a given pair of bodies occur in equal times. This definition is proved to be real by the fact, that the comparative measurements of forces made in different intervals of time are consistent with each other, which would not be the case if the definition were fantastic.
-Let the “mass” of a body mean a quantity inversely proportional to the change of velocity impressed on that body in a given time by a given force. This definition is proved to be real by the fact that the ratio of the masses of two given bodies is found experimentally to be always the same, when those masses are compared by means of the velocities impressed on them by different forces, and in different times; and is also the same, whether each of the masses is measured as a whole or as the sum of a set of parts.
Assuming those definitions as merely verbal, without reference to their reality, the laws of motion take the form of verbal truisms; but when experiment and observation inform us that permanent relations exist amongst real bodies and real events corresponding to the definitions, those apparent truisms become statements of fact.
One of the chief objects of mathematical physics is to ascertain, by the help of experiment and observation, what physical quantities or functions are “conserved.” Such quantities or functions are, for example:
-The mass of every particle of matter, conserved at all times and under all circumstances.
-The resultant momentum of a body, or system of bodies, conserved so long as internal forces act alone.
-The resultant angular momentum of a body or system of bodies, conserved so long as internal forces act alone.
-The total energy of a body, or system of bodies, conserved so long as internal forces act alone.
-The thermo-dynamic function, conserved in a body while it neither receives nor gives out heat.
In defining such physical quantities as those, it is almost, if not quite, impossible to avoid making the definition imply the property of conservation; so that when the fact of conservation is stated, it has the form of a truism.
In conclusion, it appears to me that the making of a physical law wear the appearance of a truism, so far from being a ground of objection to the definition of a physical term, is rather a proof that such definition has been framed in strict accordance with reality.
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